Prof. Conrad mentioned in a recent answer that most of the (introductory?) books on reductive groups do not make use of scheme theory. Do any books using scheme theory actually exist? Further, are there any books that use the functor of points approach? Demazure-Gabriel's second book would have covered general group schemes in this way, but it was never written, and it's not clear whether or not it would have covered reductive groups anyway. There is a lot of material in SGA 3 using more modern machinery to study group schemes, but I'm not aware of any significant treatment of reductive groups in that book (although I haven't read very much of it).

Correction: Prof. Conrad has noted that SGA 3 does contain a significant treatment of reductive groups using modern machinery.

Even though, when I was a graduate student, I was firmly a "scheme" person, and looked upon varieties with some disdain, I realised later on that for the theory of reductive groups it's well worth setting up the basics over an alg closed field first, and then over a general field, and then over a general scheme. This is not often the case---e.g. I'm not sure what advantage there would be in defining a proper morphism of varieties and then a proper morphism of schemes---but for alg groups there is a definitely a case for it, esp if you're interested in Langlands type stuff.
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Kevin BuzzardMar 10 '10 at 8:05

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Is there a simple definition of a reductive group scheme using the functor of points approach?
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Harry GindiMar 10 '10 at 8:53

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On the way to work I thought of an analogy. When I was a grad student I was very enthusiastic about finite flat group schemes, and knew several good references that treated these gadgets over bases like spec(complete DVR) and so on. However of course I would sometimes invoke arguments from abstract group theory as a matter of course when reading and understanding proofs. So I don't know why I expected to be able to understand reductive group schemes without first understanding reductive groups over an alg closed field base.
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Kevin BuzzardMar 10 '10 at 13:09

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@fpqc: even in SGA3 the definition rests on a criterion with geometric fibers. Reductivity is fundamentally a geometric notion: over an imperfect field $k$ we cannot detect it using just $k$-subgroups. That is the whole point of the pseudo-reductive stuff. Functor of points is a convenient notion, even powerful for working with group objects and related constructions, but not everything can be shoehorned into it. Can you define (in a useful non-tautologous way...) the dimension of a finite type scheme over a field or quasi-compactness or irreducibility using functor of points?
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BCnrdMar 11 '10 at 4:46

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@fpqc: that defn of q-c for alg spaces does not make sense for "general" functors on rings (cf. functorial criteria for lfp, smooth, proper). Since alg spaces are functors by defn (no loc. ringed space), it is a tautology that any definition for alg spaces is literally expressed in terms of its "functor of points" since that's all it is! The crux is whether the defn/criterion is meaningful for a functor not known to be an algebraic space. For q-c (and irred) the answer is "no". I put "useful non-tautologous way" in my previous comment to head off a reply as you wrote.
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BCnrdMar 13 '10 at 18:02

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Personally, I find the "classical" books (Borel, Humphreys, Springer) unpleasant to read because they work in the wrong category, namely, that of reduced algebraic group schemes rather than all algebraic group schemes. In that category, the isomorphism theorems in group theory fail, so you never know what is true. For example, the map $H/H\cap N\rightarrow HN/N$ needn't be an isomorphism (take $G=GL_{p}$, $H=SL_{p}$, $N=\mathbb{G}_{m}$ embedded diagonally). Moreover, since the terminology they use goes back to Weil's Foundations, there are strange statements like "the kernel of a homomorphism of algebraic groups defined over $k$ need not be defined over $k$". Also I don't agree with Brian that if you don't know descent theory, EGA, etc. then you don't "know scheme theory well enough to be asking for a scheme-theoretic treatment'.

Which explains why I've been working on a book whose goal is to allow people to learn the theory of algebraic group schemes (including the structure of reductive algebraic group schemes) without first reading the classical books and with only the minimum of prerequisites (for what's currently available, see my website under course notes). In a sense, my aim is to complete what Waterhouse started with his book.

So my answer to the question is, no, there is no such book, but I'm working on it....

Brilliant! Is there any way I can be "kept posted" on that book?
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Harry GindiMar 11 '10 at 1:30

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@Jim: When learning these things on my own, the only way I could make scheme-theoretic versions of the proofs of various things was to use what I knew from EGA, descent theory, etc.. So my impression was that without a certain amount of "post-Hartshorne" algebraic geometry technique already known, an attempt to do this stuff scheme-theoretically in a clean way would fail. I'd be happy to be proved wrong! Or even better, for there to be an alg. gps. book that, like with "Neron Models", develops the necessary techniques and then applies it to do interesting things.
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BCnrdMar 11 '10 at 4:34

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Straight from the author's website: May 2010. New version of Algebraic Groups, Lie Groups, and their Arithmetic Subgroups
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AnonymousMar 11 '10 at 4:35

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@Brian: I agree that if you try to directly transfer the proof in the smooth case to the nonsmooth case, you can sometimes run into some very heavy scheme theory, but there are also elementary proofs using Hopf algebras. I learnt this from Waterhouse's book. As Serre pointed out, Hopf algebra proofs don't illuminate, but my strategy is to sketch the geometric argument and write out the Hopf algebra argument (when necessary). I'm only doing things over fields (or rings, when it's just as easy).
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JS MilneMar 11 '10 at 5:14

Oh my goodness, SGA3 is an absolutely fundamental reference on the theory of reductive groups. The significance of its treatment is tremendous. But it freely assumes familiarity with the theory over an algebraically closed field. I see that as no big deal. It is like learning some basics about varieties before schemes: totally reasonable, almost absurd to do otherwise (in terms of understanding where the ideas come from, having experience with real examples on which to test one's knowledge of subtleties, etc.)

I think it is a perfectly good thing to first learn the theory over fields by reading one of these more "classical" books, since all of the serious work takes place there and the relativization brings in other tools to bootstrap from the field case (i.e., one cannot do anything serious without first setting up a good theory over fields, just like real theorems about abelian schemes rely on first doing the theory of abelian varieties). That being said, when learning the theory over fields from these "classical" books, those who wish for a scheme-theoretic treatment can just re-interpret most of the proofs in terms of schemes if one wishes to do so (using knowledge of topics from descent theory, EGA, etc.) If that is too hard, it means one doesn't know scheme theory well enough to be asking for a scheme-theoretic treatment. I should note that even Jantzen assumes familiarity with a fair amount of the theory over fields. There is nothing wrong with that, in my opinion.

Oh, I guess I misunderstood your answer in the other question. Thanks for the information!
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Harry GindiMar 9 '10 at 23:54

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I really don't think of the older books as "classical", but instead as a response to particular needs at a particular time. The books emerged from a period in the late 1960s when many people wanted to assimilate and develop the ideas in the Chevalley classification seminar. Algebraic geometry at the time was very much in flux, with the approaches of Weil and Chevalley rapidly giving way. Borel's lectures at Columbia were written up by Bass and later turned into a sort of textbook by me, followed by Springer's and then an expanded book by Borel himself. All very ad hoc.
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Jim HumphreysMar 13 '10 at 20:45